Handling uncertainty in economic nonlinear model predictive control: A comparative case study

In the last years, the use of an economic cost function for model predictive control (MPC) has been widely discussed in the literature. The main motivation for this choice is that often the real goal of control is to maximize the profit or the efficiency of a certain system, rather than tracking a predefined set-point as done in the typical MPC approaches, which can be even counter-productive. Since the economic optimal operation of a system resulting from the application of an economic model predictive control approach drives the system to the constraints, the explicit consideration of the uncertainties becomes crucial in order to avoid constraint violations. Although robust MPC has been studied during the past years, little attention has yet been devoted to this topic in the context of economic nonlinear model predictive control, especially when analyzing the performance of the different MPC approaches. In this work, we present the use of multi-stage scenario-based nonlinear model predictive control as a promising strategy to deal with uncertainties in the context of economic NMPC. We make a comparison based on simulations of the advantages of the proposed approach with an open-loop NMPC controller in which no feedback is introduced in the prediction and with an NMPC controller which optimizes over affine control policies. The approach is efficiently implemented using CasADi, which makes it possible to achieve real-time computations for an industrial batch polymerization reactor model provided by BASF SE. Finally, a novel algorithm inspired by tube-based MPC is proposed in order to achieve a trade-off between the variability of the controlled system and the economic performance under uncertainty. Simulations results show that a closed-loop approach for robust NMPC increases the performance and that enforcing low variability under uncertainty of the controlled system might result in a big performance loss.     

A robust distributed model predictive control algorithm

Although distributed model predictive control (DMPC) has received significant attention in the literature, the robustness of DMPC with respect to model errors has not been explicitly addressed. In this paper, a novel online algorithm that deals explicitly with model errors for DMPC is proposed. The algorithm requires decomposing the entire system into N subsystems and solving N convex optimization problems to minimize an upper bound on a robust performance objective by using a time-varying state-feedback controller for each subsystem. Simulations examples were considered to illustrate the application of the proposed method.     

A hybrid Gaussian process approach to robust economic model predictive control

This paper proposes a hybrid Gaussian process (GP) approach to robust economic model predictive control under unknown future disturbances in order to reduce the conservatism of the controller. The proposed hybrid GP is a combination of two well-known methods, namely, kernel composition and nonlinear auto-regressive. A switching mechanism is employed to select one of these methods for disturbance prediction after analyzing the prediction outcomes. The hybrid GP is intended to detect not only patterns but also unexpected behaviors in the unknown disturbances by using past disturbance measurements. A novel forgetting factor concept is also utilized in the hybrid GP, giving less weight to older measurements, in order to increase prediction accuracy based on recent disturbances values. The detected disturbance information is used to reduce prediction uncertainty in economic model predictive controllers systematically. The simulation results show that the proposed method can improve the overall performance of an economic model predictive controller compared to other GP-based methods in cases when disturbances have discernible patterns.     

Realization of robust nonlinear model predictive control by offline optimisation

This paper describes a new method for increasing the computational efficiency of nonlinear robust model-based predictive control. It is based on the application of neuro-fuzzy networks and improves the computation efficiency by arranging the online optimisation to be done offline. The offline optimisation is realized by offline training a neuro-fuzzy network, consisting of zero-order T–S fuzzy rules, which is designed to approximate the input–output relationship of a robust model-based predictive controller. The design and the training of the neuro-fuzzy network are described, and the corresponding control algorithm is developed. Experiment results performed on the temperature control loop of an experimental air-handling unit (AHU) demonstrate the effectiveness of this approach.     

Fast economic model predictive control based on NLP-sensitivities

This study adapts the advanced step NMPC framework to Economic NMPC. Here, sufficient conditions for nominal stability are derived for NMPC controllers that incorporate economic stage costs with appropriate regularization. To guarantee these conditions, we derive a constructive strategy to calculate the regularization term directly. Moreover, we extend the sensitivity components in the advanced step NMPC framework to consider a rigorous path-following algorithm. This approach accounts for active set changes and allows much weaker constraint qualifications. Moreover, using an ℓ₁ formulation of the NMPC problem satisfies these constraint qualifications and allows more reliable solution of the moving horizon optimization problem, even in the presence of noise. Finally, all of these concepts are demonstrated on a detailed case study with a continuously stirred tank reactor.     

A switching robust model predictive control approach for nonlinear systems

This work considers enhancing the stability and improving the economic performance of nonlinear model predictive control in the presence of disturbances or model uncertainties. First, a robust control Lyapunov function (RCLF)-based predictive control strategy is proposed. Second, the approximate dynamic programming (ADP) is employed to further improve regulation performance. Finally, the ADP and RCLF-MPC are combined to provide a switching control scheme, which is illustrated on a CSTR example to show its effectiveness.     

A new robust model predictive control method I: theory and computation

The success of the single-model MPC (SMPC) controller depends on the accuracy of the process model. Modeling errors cause sub-optimal control performance and may cause the control system to become closed-loop unstable. The goal of this paper is to examine the control performance of the robust MPC (RMPC) method proposed by Wang and Rawlings [34] on several illustrative examples. In this paper, we show the RMPC method successfully controls systems with time-varying uncertainties in the process gain, time constant and time delay and achieves offset-free non-zero set point tracking and non-zero disturbance rejection subject to input and output constraints.     

Interactions between control and process design under economic model predictive control

Economic model predictive control (EMPC) is a model-based control scheme that integrates process control and economic optimization, which can potentially allow for time-varying operating policies to maximize economic performance. The manner in which an EMPC operates a process to optimize economics depends on the process dynamics, which are fixed by the process design. This raises the question of how process and EMPC designs interact. Works which have addressed process and control design interactions for steady-state operation have sought to simultaneously develop process designs and control law parameters to find the most profitable way to operate a process that is able to prevent process constraints from being violated and to optimize capital costs in the presence of disturbances. Because EMPC has the potential to operate a process in a transient fashion, this work first focuses on how EMPC and process design interact in the absence of disturbances. Using small-scale process examples, we seek to understand the fundamental nature of the interactions between EMPC and process design, including how these interactions can impact computational complexity of the controller and the design procedure. We subsequently utilize the insights gained to suggest controller design variables which might be considered as decision variables for a simultaneous process and control design problem when disturbances are considered.     

Adaptive horizon economic nonlinear model predictive control

In this paper, we present a computationally efficient economic NMPC formulation, where we propose to adaptively update the length of the prediction horizon in order to reduce the problem size. This is based on approximating an infinite horizon economic NMPC problem with a finite horizon optimal control problem with terminal region of attraction to the optimal equilibrium point. Using the nonlinear programming (NLP) sensitivity calculations, the minimum length of the prediction horizon required to reach this terminal region is determined. We show that the proposed adaptive horizon economic NMPC (AH-ENMPC) has comparable performance to standard economic NMPC (ENMPC). We also show that the proposed adaptive horizon economic NMPC framework is nominally stable. Two benchmark examples demonstrate that the proposed adaptive horizon economic NMPC provides similar performance as the standard economic NMPC with significantly less computation time.     

Convergence in economic model predictive control with average constraints

In this paper, we thoroughly investigate various aspects of economic model predictive control with average constraints, i.e., constraints on average values of state and input variables. In particular, we first show that a certain time-varying output constraint has to be included into the MPC problem formulation in order to ensure fulfillment of these average constraints. Optimizing a general (possibly economic) performance criterion may result in a non-converging behavior of the corresponding closed-loop system. While such a behavior might be acceptable in some cases, it may be undesirable for other types of applications. Hence as a second contribution, we provide a Lyapunov-like analysis to conclude that indeed asymptotic convergence to the optimal steady-state follows if the system satisfies a certain dissipativity condition. Finally, for the case that this dissipativity property is not satisfied but still a convergent behavior of the closed-loop is required, we examine two different methods how convergence can be enforced within an economic MPC setup by imposing additional average constraints on the system. In the first method, an additional average constraint is defined which results in the system being dissipative, while the second consists of imposing an additional even zero-moment average constraint. We illustrate our results with various examples.     

基于多面体不变集的离线鲁棒预测控制器综合

提出一种基于多面体不变集的离线鲁棒预测控制器综合算法.该算法离线确定一组反馈控制律及其对应的不变集,在线控制时根据当前状态所处的位置选择相应的控制律,不仅扩大了初始可行域,还能在一定程度上改善控制性能.仿真结果表明了采用多面体不变集的优越性.     

On real-time robust model predictive control

High-speed applications impose a hard real-time constraint on the solution of a model predictive control (MPC) problem, which generally prevents the computation of the optimal control input. As a result, in most MPC implementations guarantees on feasibility and stability are sacrificed in order to achieve a real-time setting. In this paper we develop a real-time MPC approach for linear systems that provides these guarantees for arbitrary time constraints, allowing one to trade off computation time vs. performance. Stability is guaranteed by means of a constraint, enforcing that the resulting suboptimal MPC cost is a Lyapunov function. The key is then to guarantee feasibility in real-time, which is achieved by the proposed algorithm through a warm-starting technique in combination with robust MPC design. We address both regulation and tracking of piecewise constant references. As a main contribution of this paper, a new warm-start procedure together with a Lyapunov function for real-time tracking is presented. In addition to providing strong theoretical guarantees, the proposed method can be implemented at high sampling rates. Simulation examples demonstrate the effectiveness of the real-time scheme and show that computation times in the millisecond range can be achieved.     

Periodic optimal control of nonlinear constrained systems using economic model predictive control

In this paper, we consider the problem of periodic optimal control of nonlinear systems subject to online changing and periodically time-varying economic performance measures using model predictive control (MPC). The proposed economic MPC scheme uses an online optimized artificial periodic orbit to ensure recursive feasibility and constraint satisfaction despite unpredictable changes in the economic performance index. We demonstrate that the direct extension of existing methods to periodic orbits does not necessarily yield the desirable closed-loop economic performance. Instead, we carefully revise the constraints on the artificial trajectory, which ensures that the closed-loop average performance is no worse than a locally optimal periodic orbit. In the special case that the prediction horizon is set to zero, the proposed scheme is a modified version of recent publications using periodicity constraints, with the important difference that the resulting closed loop has more degrees of freedom which are vital to ensure convergence to an optimal periodic orbit. In addition, we detail a tailored offline computation of suitable terminal ingredients, which are both theoretically and practically beneficial for closed-loop performance improvement. Finally, we demonstrate the practicality and performance improvements of the proposed approach on benchmark examples.     

On the explicit dead-time compensation for robust model predictive control

Computational simplicity is one of the most important aspects to take into account in robust model predictive control (MPC). In dead-time processes, it is common to use an augmented state-space representation in order to apply robust MPC strategies but, this procedure may affect computational aspects. In this paper, explicit dead-time compensation will be used to avoid augmented representation. This technique will be analyzed in terms of robust stability and constraint satisfaction for discrete-time linear systems. The results of this discussion will be applied to a robust tube-based MPC strategy which is able to guarantee robust stability and constraint satisfaction of a dead-time system by considering a prediction model without dead-time. Moreover, taking advantage of the proposed scheme, the robust MPC will be particularized for first-order plus dead-time models which simplifies significantly controller synthesis. The proposed dead-time compensation method will be applied to different robust MPC strategies in two case studies: (i) a simulated quadruple-tank system, and (ii) an experimental scaled laboratory heater process.     

Transient average constraints in economic model predictive control

In this paper, an economic model predictive control algorithm is proposed which ensures satisfaction of transient average constraints, i.e., constraints on input and state variables averaged over some finite time period. We believe that this stricter form of average constraints (compared to previously proposed asymptotic average constraints) is of independent interest in various applications such as the operation of a chemical reactor, where e.g. the amount of inflow or the heat flux during some fixed period of time must not exceed a certain value. Besides guaranteeing fulfillment of transient average constraints for the closed-loop system, we show that closed-loop average performance bounds and convergence results established in the setting of asymptotic average constraints also hold in case of transient average constraints. Furthermore, we illustrate our results with a chemical reactor example.     

A tutorial review of economic model predictive control methods

An overview of the recent results on economic model predictive control (EMPC) is presented and discussed addressing both closed-loop stability and performance for nonlinear systems. A chemical process example is used to provide a demonstration of a few of the various approaches. The paper concludes with a brief discussion of the current status of EMPC and future research directions to promote and stimulate further research potential in this area.     

Implications of discretization on dissipativity and economic model predictive control

Economic model predictive control, where a generic cost is employed as the objective function to be minimized, has recently gained much attention in model predictive control literature. Stability proof of the resulting closed-loop system is often based on strict dissipativity of the system with respect to the objective function. In this paper, starting with a continuous-time setup, we consider the ‘discretize then optimize’ approach to solving continuous-time optimal control problems and investigate the effect of the discretization process on the closed-loop system. We show that while the continuous-time system may be strictly dissipative with respect to the objective function, it is possible that the resulting closed-loop system is unstable if the discrete-approximation of the continuous-time optimal control problem is not properly set up. We use a popular example from the economic MPC literature to illustrate our results.     

Least-restrictive robust periodic model predictive control applied to room temperature regulation

State-feedback model predictive control (MPC) of constrained discrete-time periodic affine systems is considered. The periodic systems’ states and inputs are subject to periodically time-dependent, hard, polyhedral constraints. Disturbances are additive, bounded and subject to periodically time-dependent bounds. The objective is to design MPC laws that robustly enforce constraint satisfaction in a manner that is least-restrictive, i.e., have the largest possible domain. The proposed design method is demonstrated on a building climate control example. The proposed method is directly applicable to time-invariant MPC.